A spacecraft is traveling to the Moon and uses a lunar gravity assist maneuver to increase its speed. Determine the speed of the spacecraft after the slingshot if its initial speed was 10,000 m/s and it approached the Moon at a distance of 1,000 km.
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To determine the speed of the spacecraft after the lunar gravity assist maneuver, we can use the conservation of energy and momentum.
First, let's convert 1,000 km to meters:
1,000 km = 1,000,000 m
The spacecraft's initial speed is given as 10,000 m/s.
Assuming there is no loss or gain of energy during the maneuver, the total mechanical energy of the spacecraft remains constant. Therefore, the sum of the kinetic energy and the gravitational potential energy remains constant.
Initial kinetic energy = (1/2)mv^2, where m is the mass of the spacecraft and v is the initial speed.
Final kinetic energy = (1/2)mv'^2, where v' is the final speed.
Initial gravitational potential energy = (-GMm)/r, where G is the gravitational constant, M is the mass of the Moon, m is the mass of the spacecraft, and r is the distance between the center of the Moon and the spacecraft.
Final gravitational potential energy = (-GMm)/R, where R is the distance between the center of the Earth and the spacecraft after the maneuver.
Since the spacecraft approaches the Moon, initial gravitational potential energy is negative and final gravitational potential energy is also negative.
Therefore, we have:
(1/2)mv^2 - GMm/r = (1/2)mv'^2 - GMm/R
Since the gravitational force on the spacecraft is primarily due to the Moon, we have:
GMm/r^2 = mv^2/r
Rearrange the above equation to solve for m:
m = (v^2 * r) / (G)
Substitute this value of m into the equation for gravitational potential energy:
(1/2) * (v^2 * r) / (G) * v^2 - GM * (v^2 * r) / (G) = (1/2)mv'^2 - GMm/R
Cancel out common terms and rearrange the equation to solve for v':
(v^2 * r) - GM * (v^2 * r) = (v'^2 * R) - GM * (v^2 * r)
Rearrange the equation to solve for v':
v'^2 * R = v^2 * (r - GM * r)
v' = √[(v^2 * (r - GM * r)) / R]
Substituting the given values:
v' = √[(10,000^2 * (1,000,000 - (6.674 * 10^-11 * 7.342 × 10^22 * (1,000,000 * 10^3)))) / (1,000,000 * 10^3))]
Calculating this equation will give us the final speed of the spacecraft after the lunar gravity assist maneuver.
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